# Why is Mathematics Possible: Tim Gowers’s Take on the Matter

In a previous post I mentioned the question of why is mathematics possible. Among the interesting comments to the post, here is a comment by Tim Gowers:

“Maybe the following would be a way of rephrasing your question. We know that undecidability results don’t show that mathematics is impossible, since we are interested in a tiny fraction of mathematical statements, and in practice only in a tiny fraction of possible proofs (roughly speaking, the comprehensible ones). But why is it that these two classes match up so well? Why is it that nice mathematical statements so often have proofs that are of the kind that we are able to discover?

# Why is mathematics possible?

## Spectacular advances in number theory

Last weeks we heard about two spectacular results in number theory.  As announced in Nature, Yitang Zhang proved that there are infinitely many pairs of consecutive primes $(p_n, p_{n+1})$ which are at most 70 million apart! This is a sensational achievement. Pushing 70 million to 2 will settle the ancient conjecture on twin primes, but this is already an extremely amazing breakthrough.  An earlier breakthrough came in 2005 when Daniel Goldston, János Pintz, and Cem Yıldırım proved that the gaps between consecutive primes $p_{n+1}-p_n$ is infinitely often smaller than $\sqrt {\log p_n} \log \log ^2 p_n$.

Update: A description of Zhang’s work and a link to the paper can be found on Emmanuel Kowalski’s bloog Further update: A description of Zhang’s work and related questions and results can be found now in Terry Tao’s blog. Terry Tao also proposed a new polymath project aimed to reading Zhang’s paper and attempting to improve the bounds.

Harald Helfgott proved that every integer is the sum of three primes.  Here the story starts with Vinogradov who proved it for sufficiently large integers, but pushing down what “sufficiently large” is, and pushing up the computerized methods needed to take care of “small” integers required much work and ingenuity.

## Why is Mathematics possible?

The recent news, and a little exchange of views I had with Boaz Barak, bring us back to the question: “Why is mathematics possible?” This is an old question that David Kazhdan outlined in a lovely 1999 essay “Reflection on the development of mathematics in the twentieth century.” The point (from modern view) is this: We know that mathematical statements can, in general, be undecidable.  We also know that a proof for a short mathematical statement can be extremely long. And we also know that even if a mathematical statement admits a short proof, finding such a proof can be computationally intractable. Given all that, what are the reasons that mathematics is at all possible?

It is popular to associate “human creativity” with an answer. The problem with incorrect (or, at least, incomplete) answers is that they often represent missed opportunities for better answers. I think that for the question “why is mathematics possible” there are opportunities (even using computational complexity thinking) to offer better answers.

# Polynomial Hirsch Conjecture 5: Abstractions and Counterexamples.

This is the 5th research thread of polymath3 studying the polynomial Hirsch conjecture. As you may remember, we are mainly interested in an abstract form of the problem about families of sets. (And a related version about families of multisets.)

The 4th research thread was, in my opinion, fruitful. An interesting further abstraction was offered and for this abstraction a counterexample was found. I will review these developments below.

There are several reasons why the positive direction is more tempting than the negative one. (And as usual, it does not make much of a difference which direction you study. The practices for trying to prove a statement and trying to disprove it are quite similar.) But perhaps we should try to make also some more pointed attempts towards counterexamples?

Over the years, I devoted much effort including a few desperate attempts to try to come up with counterexamples. (For a slightly less abstract version than that of EHRR.) I tried to base one on the Towers of Hanoi game. One can translate the positions of the game into a graph labelled by subsets. But the diameter is exponential! So maybe there is a way to change the “ground set”? I did not find any. I even tried to look at games (in game stores!) where the player is required to move from one position to another to see if this leads to an interesting abstract example. These were, while romantic, very long shots.

Two more things: First, I enjoyed meeting in Lausanne for the first time Freidrich Eisenbrand, Nicolai Hahnle, and Thomas Rothvoss. (EHR of EHRR.) Second, Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick proved (mildly) subexponential lower bounds for certain randomized pivot steps for the simplex algorithm. We discussed it in this post.  The underlying polytopes in their examples are combinatorial cubes. So this has no direct bearing on our problem. (But it is interesting to see if geometric or abstract examples coming from more general games of the type they consider may be relevant.)

So let me summarize PHC4 excitements and, as usual, if I missed something please add it.

# Polymath3: Polynomial Hirsch Conjecture 4

So where are we? I guess we are trying all sorts of things, and perhaps we should try even more things. I find it very difficult to choose the more promising ideas, directions and comments as Tim Gowers and Terry Tao did so effectively in Polymath 1,4 and 5.  Maybe this part of the moderator duty can also be outsourced. If you want to point out an idea that you find promising, even if it is your own idea, please, please do.

This post has three parts. 1) Around Nicolai’s conjecture; 1) Improving the upper bounds based on the original method; 3) How to find super-polynomial constructions? Continue reading

# Polymath3 : Polynomial Hirsch Conjecture 3

Here is the third research thread for the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem we discuss. Even more important, to think about the problem either in the directions suggested by others or on their own. Participants who follow the project and think about the issues without adding remarks are valuable.

The combinatorial problem is simple to state and also everything that we know about it is rather simple. At this stage joining the project should be easy.

Let me try to describe (without attemting to be complete) one main direction that we discuss. This direction started with the very first comment we had by Nicolai.

Please do not hesitate to repeat an idea raised by yourself or by other if you think it can be useful.

Let $f^*(d,n)$ be the largest number of disjoint families $F_1, F_2, ..., F_t$ of degree d monomials in the variables $x_1,\dots,x_n$ such that

(*) for i < j < k, whenever $m_1 \in F_i$ and $m_2 \in F_k$, then there exists a monomial $u \in F_j$ such that $gcd(m_1, m_2) | u$.

### Nicolai’s conjecture:

$f^*(d,n)=d(n-1)+1$.

The example that supports this conjecture consists of families with a single monomial in every family.

The monomials are

$x_1^d,$

$x_1^{d-1}x_2$,

$\dots$,

$x_2^d$,

$x_2^{d-1}x_3$,

$\dots$,

$x_n^d$.

There are other examples that achieve the same bound. The bound can be achieved by families whose union include all monomials, and for such families the conjecture is correct.

### The case d=3.

An upper bound by EHRR (that can be extended to monomials) following works of Barnette and Larman on polytopes is $f^*(d,n) \le 2^{d-1}n$. For degree 3 monomials we have a gap

$3n-2\le f^*(3,n) \le 4n-1$.

It may be the case that understanding the situation for $d=3$ is the key for the whole problem.

There is another example achieving the lower bound that Terry found

$F_i := \{ \{a,b,c\}: a+b+c = i+2 \}$ $i=1,2,\dots 3n-2$

### Various approaches to the conjecture

Several approaches to the cojecture were proposed. Using clever reccurence relations, finding useful ordering, applying the method of compression, and algebraic methods. In a series of remarks Tim is trying to prove Nicolai’s conjecture. An encouraging sign is that both examples of Nicolai, Klas, and Terry come up naturally. One way to help the project at this stage would be to try to enter Tim’s mind and find ways to help him “push the car”. In any case, if Nicolai’s conjecture is correct I see no reason why it shouldn’t have a simple proof (of course we will be happy with long proofs as well).

### Constructions

Something that is also on the back of our minds is the idea to find examples that are inspired from the upper bound proofs. We do not know yet what direction is going to prevail so it is useful to remember that every proof of a weaker result and every difficulty in attempts to proof the hoped-for result can give some ideas for disproving what we are trying to prove.

Some preliminary attempts were made to examine what are the properties of examples for d=3 which will come close to the 4n bound. It may also be the case that counterexamples to Nicolai’s conjecture can be found for rather small values of n and d.

Two polls:

# Polymath 3: The Polynomial Hirsch Conjecture 2

Here we start the second research thread about the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem. The combinatorial problem looks simple and also everything that we know about it is rather simple: At this stage joining the project should be very easy. If you have an idea (and certainly a question or a request,) please don’t feel necessary to read all earlier comments to see if it is already there.

In the first post we described the combinatorial problem: Finding the largest possible number f(n) of disjoint families of subsets from an n-element set which satisfy a certain simple property (*).We denote by f(d,n) the largest possible number of families satisfying (*) of d-subsets from {1,2,…,n}.

The two principle questions we ask are:

Can the upper bounds be improved?

and

Can the lower bounds be improved?

What are the places that the upper bound argument is wasteful and how can we improve it? Can randomness help for constructions? How does a family for which the upper bound argument is rather sharp will look like?

We are also interested in the situation for small values of n and for small values of d. In particular, what is f(3,n)? Extending the problem to multisets (or monomials) instead of sets may be fruitful since there is a proposed suggestion for an answer.

# Polymath 3: Polynomial Hirsch Conjecture

I would like to start here a research thread of the long-promised Polymath3 on the polynomial Hirsch conjecture.

I propose to try to solve the following purely combinatorial problem.

Consider t disjoint families of subsets of {1,2,…,n}, $F_1, F_2, ..., F_t$.

Suppose that

(*) For every $i, and every $S \in F_i$ and $T \in F_k$, there is $R\in F_j$ which contains $S\cap T$.

The basic question is: How large can t  be???

(When we say that the families are disjoint we mean that there is no set that belongs to two families. The sets in a single family need not be disjoint.)

In a recent post I showed the very simple argument for an upper bound $n^{\log n+1}$. The major question is if there is a polynomial upper bound. I will repeat the argument below the dividing line and explain the connections between a few versions.

A polynomial upper bound for $f(n)$ will imply a polynomial (in $n$) upper bound for the diameter of graphs of polytopes with $n$ facets. So the task we face is either to prove such a polynomial upper bound or give an example where $t$ is superpolynomial.

The abstract setting is taken from the paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss. They gave an example that $f(n)$ can be quadratic.

Remark: The comments for this post will serve both the research thread and for discussions. I suggested to concentrate on a rather focused problem but other directions/suggestions are welcome as well.